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MATH 163, Spring 2001 Due Date: Name(s):
Project 4.5b: Graphing Functions
Objective
To investigate how we can relate where a function f and its first and second derivatives are positive,
negative, zero, and do not exist, and the graph of f.
Narrative
The author provides one mechanism for gathering information to be used in graphing a function. In this
project we present another.
To record where a function f and its first and second
derivatives f = f1 and f = f2, respectively, are positive,
negative, zero, and do not exist, we use three "recording
strips" below the graph of f as illustrated in the figure at
the right; here f(x) = (x4
- 4x3
)/10. (See Example 6,
p. 245 of the text.) Study this figure carefully. Note how
the vertical bars are drawn where f , f , and f are 0 or
do not exist, and the spaces between the vertical bars are
labelled + or - depending on the sign of f , f , and f
over those intervals.
Tasks
1. Type the command lines below into Maple in the order in which they are listed. They initialize Maple
for this project.
> # Project 4.5b: Graphing Functions
> restart;
> with(plots):
2. Type the command lines below into Maple in the order in which they are listed. They produce a graph
of the function f(x) = 4x/(1 + x2
), and three recording strips below the graph of f.
> # Task 2
> f := x -> 4*x/(1+x^2);
> plot0 := plot({-6,-5,-4,-3,0},x=-6..6,y=-6..3):
> plot1 := textplot({[-6,-3.5,'f'],[-6,-4.5,'f1'],[-6,-5.5,'f2']}):
> plot2 := plot(f(x),x=-6..6):
> display({plot0,plot1,plot2});
3. a) Type the command lines below into Maple in the order in which they are listed. They determine
where f(x) = 3x4
- 16x3
+ 18x2
and its first and second derivatives are positive, negative, zero, and do
not exist; this information will be used later in this project.
> # Task 3
> f := x -> 3*x^4-16*x^3+18*x^2;
> evalf(solve(f(x)=0,x));
> f1 := D(f);
> evalf(solve(f1(x)=0,x));
> f2 := D(f1);
> evalf(solve(f2(x)=0,x));
b) Type the command line below into Maple. It produces an empty graph and three recording strips.
> display({plot0,plot1});
At this time, make a hard-copy of your input and Maple's responses. Then, ...
4. Fill in the recording strips on the graphic you produced in Task 2 using the graph of f in that graphic
as a guide.
5. a) Fill in the recording strips on the graphic you produced in Task 3(b) using the information you
computed in Task 3(a) as a guide.
b) Use the information you recorded in (a) to sketch the graph of f in the space provided.
Comments
To find the intercepts and vertical asymptotes of the graph of f we need to find the values of x for which
f(x) = 0 and f(x) does not exist, to find the (possible) critical values of f we need to find the values of x
for which f (x) = 0 and f (x) does not exist, and to find the (possible) inflection points of the graph of f,
we need to find the values of x for which f (x) = 0 and f (x) does not exist. Since for every polynomial
function f the values of f(x), f (x), and f (x) exist for all x, in the above problem we only had to use
information about where f(x) = 0, f (x) = 0, and f (x) = 0. If, for a function in a more general class of
functions, f(x), f (x), and/or f (x) did not exist for certain values of x, we would have to find these values
and label vertical bars in recording strips with . (See Extra Project 4.5c.)